Why does the ground have water 4 feet down? Is it because the elevation is e.g. 5 feet lower than other lots? If so, that might be an unsurprising conclusion, that he is not allowed to build a basement.

If it is the same elevation, maybe it is because a rainy winter raised the water level. In that case, maybe in late summer the water level would drop enough to allow building. The Journal of Light Construction had an article for contractors on how to construct a water-proof basement in such environments. The city/county might go for that possibility, if the water level dropped enough. Of course, building would be delayed.

Another idea is to just build a split-level house, with one part sunk 3 or 4 feet into the ground, and possibly adding a basement later, not underneath the rest of the house.

I recommend exploring these alternatives with the appropriate city/county building department.